Let f be a differentiable function on the open interval (a, b). Which of the following statements must be true?
I. f is continuous on the closed interval [a, b]
II. f is bounded on the open interval (a, b)
III. If a < a1< b1 < b, and f(a1) < 0 < f(b1), then there is a number c such that a1 < c < b, and f(c) = 0
(A) I and II only
(B) I and III only
(C) II and III only
(D) only III